Problem: Many years ago, the towns of Franklin and Chester agreed to begin posting their populations on signs just outside of their towns. They also agreed to update their signs once per year at the beginning of the year. During year $1$, Franklin's sign read " $\text{Franklin: Population of } 20{,}000$ ", while Chester's sign read " $\text{Chester: Population of } 25{,}000$ ". Each year, the populations grew. Specifically, Franklin's population grew by $5\%$ each year, and Chester's population grew by $500$ people each year. What is the first year in which Franklin's sign shows a larger number than Chester's sign? Year
The signs in year $2$ At the beginning of year $2$, the signs are updated. For Franklin's sign, we take the previous value, $20{,}000$, and add $5\%$ of the previous value—or just multiply the previous value by $1.05$, which amounts to the same thing: $\begin{aligned} &20{,}000+20{,}000\cdot0.05 \\\\ =&20{,}000\cdot(1+0.05) \\\\ =&20{,}000\cdot(1.05) \\\\ =& 21{,}000 \end{aligned}$ For Chester's sign, we take the previous value, $25{,}000$, and add $500$ : $\begin{aligned} &25{,}000+500 \\\\ =& 25{,}500 \end{aligned}$ The signs in year $3$ and beyond For year $3$ and beyond, we keep multiplying Franklin's population by $1.05$ and adding $500$ to Chester's population. Year Franklin Chester (Multiply by $1.05$ each year.) (Add $500$ each year.) $1$ $20{,}000$ $25{,}000$ $2$ $21{,}000$ $25{,}500$ $3$ $22{,}050$ $26{,}000$ $4$ $23{,}153$ $26{,}500$ $5$ $24{,}310$ $27{,}000$ $6$ $25{,}526$ $27{,}500$ $7$ $26{,}802$ $28{,}000$ $8$ $28{,}142$ $28{,}500$ $9$ $29{,}549$ $29{,}000$ Franklin's sign showed a larger population than Chester's sign for the first time in year number $9$. Notice: Franklin's population grows exponentially while Chester's population grows linearly.